Exponential order statistics and some combinatorial identities

It is known that the k-th (1≤k≤n) order statistic from the unit exponential distribution can be represented as a sum of independent exponential random variables. We present a proof of this result based on Laplace transform. Also, computing the Laplace transform of the k-th order statistic in two different ways and equating them, we derive several interesting combinatorial identities. A probabilistic interpretation of these identities and their generalizations are also given.